## Advanced Studies in Pure Mathematics

### Homomorphisms on groups of volume-preserving diffeomorphisms via fundamental groups

Tomohiko Ishida

#### Abstract

Let $M$ be a closed manifold. Polterovich constructed a linear map from the vector space of quasi-morphisms on the fundamental group $\pi_{1}(M)$ of $M$ to the space of quasi-morphisms on the identity component $\mathrm{Diff}_{\Omega}^{\infty} (M)_{0}$ of the group of volume-preserving diffeomorphisms of $M$. In this paper, the restriction $H^{1}(\pi_{1}(M); \mathbb{R}) \to H^{1}(\mathrm{Diff}_{\Omega}^{\infty} (M)_{0}; \mathbb{R})$ of the linear map is studied and its relationship with the flux homomorphism is described.

#### Article information

Dates
Revised: 22 August 2014
First available in Project Euclid: 4 October 2018

https://projecteuclid.org/ euclid.aspm/1538671777

Digital Object Identifier
doi:10.2969/aspm/07210387

Mathematical Reviews number (MathSciNet)
MR3726720

Zentralblatt MATH identifier
1388.37031

#### Citation

Ishida, Tomohiko. Homomorphisms on groups of volume-preserving diffeomorphisms via fundamental groups. Geometry, Dynamics, and Foliations 2013: In honor of Steven Hurder and Takashi Tsuboi on the occasion of their 60th birthdays, 387--393, Mathematical Society of Japan, Tokyo, Japan, 2017. doi:10.2969/aspm/07210387. https://projecteuclid.org/euclid.aspm/1538671777